Let 𝔻 = { 𝑧 ∈ ℂ ∶ |𝑧| < 1} and 𝑓: 𝔻 → ℂ be an analytic function given by the power series 𝑓(𝑧) = \(\rm \Sigma_{n=0}^\infty a_nz^n\), where 𝑎0 = 𝑎1 = 1 and 𝑎𝑛 = \(\frac{1}{2^{2n}}\) for 𝑛 ≥ 2. Consider the following statements:
𝑃: If 𝑧0 ∈ 𝔻, then 𝑓 is one-one in some neighbourhood of 𝑧0.
𝑄: If 𝐸 = { 𝑧 ∈ ℂ ∶ |𝑧| ≤ \(\frac{1}{2}\)}, then 𝑓(𝐸) is a closed subset of ℂ.
Which of the following statements is/are correct?
1
𝑃 is TRUE
2
𝑄 is TRUE
3
𝑄 is FALSE
4
𝑃 is FALSE